Polarimetric SAR IMage Analysis¶
Introduction¶
The Docker container mort/sardocker provides easy access to Python scripts for analysis of polarimetric synthetic aperture radar (polSAR) imagery. These scripts are described in Canty(2014), Image analysis, Classification and Change Detection in Remote Sensing, 3rd Revised Ed. In addition to scripts for polSAR speckle filtering, ENL estiation and change detection, the container encapsulates the command line interface of the ASF MapReady software for terrain correction and geocoding of SAR images. The user interacts with the software in an IPython notebook served from within the Docker container.
Software Installation¶
- Install Docker
- In a terminal, run the command
sudo docker run -d -p 433:8888 --name=sar -v your_image_directory:/home/imagery mort/sardocker
where your_image_directory is the path to your SAR data. - Point your browser to
localhost:433 - Click on this tutorial, or open a new notebook with New/Python 2.
Here is a listing of the main directory /home in the container. It contains the various Python and bash scripts required for preprocessing and change detection:
!ls -l /home
The /home/imagery directory contains the polametric SAR data and is shared with the host. In the present example there are 12 Radarsat-2 quadpol images in SLC (single-look complex) format along with a dem (digital elevation model). Acquistion times range from May 25, 2009 (20090525) to October 11, 2010 (20101011):
!ls -l /home/imagery | grep "_SLC$"
The images are level one SLC (single look complex format). For example, below are the contents of the image directory corresponding to acquistion date 20100426 (April 26, 2010). The four polarization combinations HH, HV,VH and VV are are stored as complex numbers in GeoTiff format:
ls -l /home/imagery/RS2_OK5491_PK71074_DK68879_FQ21_20100426_172459_HH_VV_HV_VH_SLC
The subdirectory polsarpro/T3 contains the polarimetric coherency matrix elements generated from the polarization combinations. This can be done with the PolSARpro provided as freeware by the European Space Agency (ESA). (See the discussion below in the Section on the processing chain.) The PolSARpro image files are in ENVI format:
ls -l /home/imagery/RS2_OK5491_PK71074_DK68879_FQ21_20100426_172459_HH_VV_HV_VH_SLC/polsarpro/T3
We'll return to these images later in the tutorial. But first a little theory:
Statistical Properties of polSAR Images¶
Vector and matrix representations¶
A fully polarimetric SAR measures a $2\times 2$ scattering matrix $S$ at each resolution cell on the ground. The scattering matrix relates the incident and the backscattered electric fields $E^i$ and $E^b$ according to
$$ \pmatrix{E_h^b \cr E_v^b} =\pmatrix{S_{hh} & S_{hv}\cr S_{vh} & S_{vv}}\pmatrix{E_h^i \cr E_v^i}. $$Here $E_h^{i(b)}$ and $E_v^{i(b)}$ denote the horizontal and vertical components of the incident (backscattered) oscillating electric fields directly at the target. These can be deduced from the transmitted and received radar signals via the so-called far field approximations. If both horizontally and vertically polarized radar pulses are emitted and discriminated then they determine, from the above Equation, the four complex scattering matrix elements. The per-pixel polarimetric information in the scattering matrix $S$, under the assumption of reciprocity ($S_{hv} = S_{vh}$), can then be expressed as a three-component complex vector
$$ s = \pmatrix{S_{hh}\cr \sqrt{2}S_{hv}\cr S_{vv}}, $$where the $\sqrt{2}$ ensures that the total intensity (received signal power) is consistent. It is essentially these vectors which are provided in the SLC level one data discussed above. The total intensity is referred to as the span and is the complex inner product of the vector $s$,
$$ {\rm span} = s^\top s = |S_{hh}|^2 + 2|S_{hv}|^2 + |S_{vv}|^2. $$This is a real number and the corresponding gray-scale image is called the span image. The observation vector $s$ can be shown to be a realization of a complex multivariate normal random variable. An equivalent and often preferred representation is in terms of the coherency vector
$$ k = {1\over\sqrt{2}}\pmatrix{S_{hh} + S_{vv}\cr S_{hh} - S_{vv} \cr 2S_{hv}}. $$The polarimetric signal is can also be represented by taking the complex outer product of $s$ with itself:
$$ C = s s^\top = \pmatrix{ |S_{hh}|^2 & \sqrt{2}S_{hh}S_{hv}^* & S_{hh}S_{vv}^* \cr \sqrt{2}S_{hv}S_{hh}^* & 2|S_{hv}|^2 & \sqrt{2}S_{hv}S_{vv}^* \cr S_{vv}S_{hh}^* & \sqrt{2}S_{vv}S_{hv}^* & |S_{vv}|^2 }. $$The diagonal elements of $C$ are real numbers, with span $= {\rm tr}(C)$, and the off-diagonal elements are complex. This matrix representation contains all of the information in the polarized signal.
Multi-looking¶
The matrix $C$ can be averaged over the number of looks (number of adjacent cells used to average out the effect of speckle) to give an estimate of the covariance matrix of each multi-look pixel:
$$ \bar{C} ={1\over m}\sum_{\nu=1}^m s(\nu) s(\nu)^\top = \langle s s^\top \rangle = \pmatrix{ \langle |S_{hh}|^2\rangle & \langle\sqrt{2}S_{hh}S_{hv}^*\rangle & \langle S_{hh}S_{vv}^*\rangle \cr \langle\sqrt{2} S_{hv}S_{hh}^*\rangle & \langle 2|S_{hv}|^2\rangle & \langle\sqrt{2}S_{hv}S_{vv}^*\rangle \cr \langle S_{vv}S_{hh}^*\rangle & \langle\sqrt{2}S_{vv}S_{hv}^*\rangle & \langle |S_{vv}|^2\rangle }, $$where $m$ is the number of looks. This matrix (or rather the equivalent coherency matrix $\langle k k^\top \rangle$) is generated by PolSARpro and stored in the subdirectory polsarpto/T3 which we listed in a previous cell above. Rewriting the first of the above equalities,
$$ m\bar{C} = \sum_{\nu=1}^m s(\nu) s(\nu)^\top =: x. $$This quantity $x$ is a realization of a complex sample matrix which is known to have a complex Wishart distribution with $N\times N$ covariance matrix $\Sigma$ (here $N=3$) and $m$ degrees of freedom:
$$ p_{W_c}( x) ={|x|^{(m-N)}\exp(-{\rm tr}(\Sigma^{-1} x)) \over \pi^{N(N-1)/2}|\Sigma|^{m}\prod_{i=1}^N\Gamma(m+1-i)},\quad m \ge N, $$provided that the vectors $s(\nu)$ are independent and drawn from a complex multivariate normal distribution.
Dual and single polarimetric imagery¶
The scattering vector given above corresponds to so-called full, or quad polarimetric SAR. Satellite-based SAR sensors often operate in reduced, power-saving polarization modes, emitting only one polarization and receiving two (dual polarization) or one (single polarization). The look-averaged covariance matrices are reduced in dimension correspondingly. For instance for dual polarization with horizontal transmission and horizontal and vertical reception,
$$ \bar{C} = \pmatrix{ \langle |S_{hh}|^2\rangle & \langle S_{hh}S_{hv}^*\rangle \cr \langle S_{hv}S_{hh}^*\rangle & \langle |S_{hv}|^2\rangle }, $$and, for single polarization and horizontal transmission/reception, we get simply the intensity image
$$ \bar{I} = \langle |S_{hh}|^2\rangle \quad {\rm or} \quad \bar{I} = \langle |S_{vv}|^2\rangle. $$Equivalent number of looks¶
However when multi-look averaging takes place, the observation vectors $s(\nu)$ are not completely independent and will generally be correlated somewhat. In order to account for this, the complex Wishart distribution is often parameterized with ENL (rather than $m$) degrees of freedom, where ENL is the so-called equivalent number of looks. This quantity can be estimated from the image itself.
Change Detection¶
Bitemporal data¶
The following is discussion is based on Conradsen et al (2004).
As we have seen, we can represent a pixel vector in an $m$ look-averaged polSAR image in covariance matrix format by $\bar C$, where
$$ m\bar C = x = \sum_{\nu=1}^m s(\nu) s(\nu)^\top $$is a realization of a random matrix $X$ with a complex Wishart distribution.
In order to derive a change detection procedure for two polarimetric SAR images, we formulate a statistical test. For each pixel in the two image, define the null (or no-change) simple hypothesis as
$$ H_0:\quad \Sigma_1 = \Sigma_2 = \Sigma, $$and the alternative composite hypothesis as
$$ H_1:\quad \Sigma_1 \ne \Sigma_2. $$Under $H_0$ the maximum likelihood for $\Sigma$ can be shown to be given by
$$ L(\hat\Sigma) = { |x_1|^{m-3}|x_2|^{m-3}\exp(-2m\cdot{\rm tr}(I)) \over \left({1\over 2m}\right)^{3\cdot 2m}| x_1+ x_2|^{2m}\Gamma_3(m)^2 }, $$where $I$ is the $3\times 3$ identity matrix and ${\rm tr}(I)=3$. Under $H_1$ the maximum likelihood for $\Sigma_1$ and $\Sigma_2$ is
$$ L(\hat\Sigma_1,\hat\Sigma_2) = { |x_1|^{m-3}|x_2|^{m-3}\exp(-2m\cdot{\rm tr}(I)) \over \left({1\over m}\right)^{3m}\left({1\over m}\right)^{3m} |x_1|^m |x_2|^m\Gamma_3(m)^2 } $$Then the likelihood ratio test has the critical region for rejection of the no-change hypothesis
$$ Q = {L(\hat\Sigma) \over L(\hat\Sigma_1,\hat\Sigma_2) } = 2^{6m}{ |x_1|^m |x_2|^m \over |x_1 + x_2|^{2m} } \le k. $$Finally, one can derive (Conradsen et al (2004)) the following approximation for the statistical distribution of the test statistic $Q$:
$$ {\rm prob}(-2\rho\log Q\le z) \simeq P_{\chi^2;N^2}(z) + \omega_2\left[ P_{\chi^2;N^2+4}(z) - P_{\chi^2;N^2}(z) \right], $$where $P_{\chi^2;m}(z)$ is the chi square distribution wth m degrees of freedom, $$ \rho = 1 - {2N^2-1\over 6N}\cdot{3\over 2 m}, $$
and
$$ \omega_2 = - {N^2\over 4}\cdot\left(1-{1\over \rho}\right)^2 + {N^2(N^2-1)\over 24 \rho^2}\cdot{7\over 4m^2}. $$In practice we choose a significance level $\alpha$, e.g., $\alpha = 0.01$, and decision threshold $z$ such that
$$ {\rm prob}(-2\rho\log Q\le z) = 1-\alpha $$and interpret all pixels with larger values of $-2\rho\log Q$ as change.
Multitemporal data¶
The preceding discussion generalizes in a straightforward way to a time series of $k$ images (Conradsen et al, Determining the points of change in time series of polarimetric SAR data, to be published). In the equation for the test statistic $Q$ above, the numerator consists of a product $k$ determinants $|x_1|\cdot|x_2|\cdot\dots|x_k|$ and the denominator similarly the determinant of the sum of the $k$ observations $|x_1+x_2+\dots x_k|$. The multitemporal test is referred to as the omnibus test will in general be more powerful (have a higher detection probability) for the same significance level than a bitemporal test just involving the first and last images.
A Processing Chain¶
The change detection method implies the following processing sequence in order to generate a change map from a time series of polarimetric SAR images provided at the single look complex (SLC) processing level:
First of all the multi-look polarimetric SAR images in covariance or coherency matrix format are generated from the SLC data with PolSARpro (or, if prefered, from the Sentinel 1 Toolbox also available from the European Space agency). Presently this must be done outside of the Docker container (and IPython) since the ESA software is only available in the form of a graphics interface. The coherenecy matrix has the same eigenvalues and hence the same determinant as the covariance matrix, so that the hypothesis test described above can be applied unchanged to either format. The rest of the processing takes place in the IPython notebook.
The matrix images are imported by MapReady for georeferencing. The bash script /home/mapready.sh in the main automates the procedure. MapReady will output the geocoded covariance/coherency matrix image in the form of co-registered GeoTiff files, one for each diagonal matrix element and two (real and imaginary parts) for each off-diagonal component. A python script /home/ingest.py is called automatically to combine these files to a single multi-band image in floating point format.
The ENL (equivalent number of looks) can (optionally) be estimated with the script enlml.py. A multivariate estimator is used as described by Anfinsen et al. (2009).
The bitemporal change detection algorithm is invoked with the bash script /home/wishart.sh. This script calls the Python programs /home/register.py to co-register the two images and then /home/wishart.py to perform the pixel-wise hypothesis tests. The test statistics $-2\rho\log Q$ and change probabilities ${\rm prob}(-2\rho\log Q\le z)$ are written to a two-band GeoTiff file. Additionally a change map showing changes at significance level 0.01 in red overlayed onto the span image is writen to a three-band (RGB) GeoTiff file. The multitemporal algorithm is invoked similarly with /home/omnibus.sh, which calls /home/register.py to co-register all of the images with the first in the series, and then /home/omnibus.py to perform the multitemporal mnibus test.
An Example: Radarsat-2 Quadpol Images¶
Returning now to the Radarsat-2 image acquired April 26, 2010, we will geocode it with MapReady (step 2 in the above processing chain). The bash script /home/mapready takes two arguments, the acquisition date in the format yyymmdd and the sensor (presently rs2quad or tsxdual):
!./mapready.sh 20100426 rs2quad
We see that the multi-look images were created from PolSARpro data with $4\times 3 = 12$ looks. This corresponds to a square pixel size of close to $20.5\times 20.5$ meters. The combined coherency matrix image at this resolution is stored in geotiff format.
Before we can display the geocoded image, we have to enable Matplotlib functionality within the notebook with the so-called magic command
%matplotlib inline
We will use the Python script /home/dispms.py for displaying. Here is the help:
run dispms -h
The command below will generate an RGB color composite of the three diagonal elements:
run /home/dispms -p [1,6,9] -d [300,300,1000,1000] \
-f /home/imagery/RS2_OK5491_PK71074_DK68879_FQ21_20100426_172459_HH_VV_HV_VH_SLC_MapReady/T3//polSAR.tif
The scene above was acquired over the city of Bonn, Germany (upper right hand corner with the Rhine river). The gray, featurless areas are mixed forest.
To check the number of looks, We will run the /home/enlml.py script (step 3 in the processing chain) on a spatial subset which includes a lot forested land cover. The spatial subset is entered with the -d flag as in the /home/dispms script.
Here we choose -d [800,400,500,500]:
run enlml -d [800,400,500,500] \
/home/imagery/RS2_OK5491_PK71074_DK68879_FQ21_20100426_172459_HH_VV_HV_VH_SLC_MapReady/T3//polSAR.tif
There are two modes (maxima) at about 6 and 12 looks. Here is the ENL image
run dispms -e 2 -f \
/home/imagery/RS2_OK5491_PK71074_DK68879_FQ21_20100426_172459_HH_VV_HV_VH_SLC_MapReady/T3/polSAR_enl.tif
The bright areas correspond to mixed forest (around 12 equivalent looks) and allow for the best estimate of the ENL, see Anfinsen et al. (2009). So we can conclude that the nominal and eqivalent number of looks are the same and equal to 12.
Next we geocode two more Radarsat-2 images, the ones acquired on May 20, 2010 (20100520) and June 7, 2010 (20100707):
!./mapready.sh 20100520 rs2quad
!./mapready.sh 20100707 rs2quad
Now let's perform the last processing step, polSAR change detection. The bitemporal bash script /home/wishart.sh needs five input parameters, the two acquistion times in yyymmdd format, a spatial subset, and the two ENL values (which in principle can be different). We choose first of all the April and July images:
!./wishart.sh 20100426 20100707 [400,400,1000,1000] 12.0 12.0
Here is the 1% significance level change map image generated by the above script:
run dispms -e 4 -p [1,2,3] -f \
/home/imagery/RS2_OK5491_PK71074_DK68879_FQ21_20100426_172459_HH_VV_HV_VH_SLC_MapReady/T3/wishart(20100426-20100707)_cmap.tif
Over the interval late April - early July separating the two acquisitions the significant changes are mostly in the agricultural areas near center right and upper left. Barge movements on the Rhine river are clearly evident. Zooming in on the upper left hand corner we can see a flooded sand quarry pit with two dredging arms that are in continual motion, giving rise to significant change signals.
run dispms -e 4 -p [1,2,3] -d [1,1,400,400] -f \
/home/imagery/RS2_OK5491_PK71074_DK68879_FQ21_20100426_172459_HH_VV_HV_VH_SLC_MapReady/T3/wishart(20100426-20100707)_cmap.tif
Finally, we'll run the omnibus test statistic on all three images. The script requires the acquisition times, the spatial subset, the number of looks and the desired significance level:
!/home/omnibus.sh 20100426 20100520 20100707 [400,400,1000,1000] 12 0.01
We will display the bitemporal and multitemporal change maps side-by-side:
run /home/dispms -e 4 -p [1,2,3] -E 4 -P [1,2,3] -d [1,1,400,400] -D [1,1,400,400] \
-f /home/imagery/RS2_OK5491_PK71074_DK68879_FQ21_20100426_172459_HH_VV_HV_VH_SLC_MapReady/T3/wishart(20100426-20100707)_cmap.tif \
-F /home/imagery/RS2_OK5491_PK71074_DK68879_FQ21_20100426_172459_HH_VV_HV_VH_SLC_MapReady/T3/omnibus(20100426-1-20100707)_cmap.tif
The multitemporal test (right hand image) detects more changes than the bitemporal test, at the same 1% significance level.
Speckle Filtering (The gamma-MAP filter)¶
Let us represent an $m$ look-averaged SAR intensity image by the random variable $G$ with mean $\langle G\rangle=x$, where $x$ is the underlying signal. Then ${\rm var}(G) = x^2/m$ and $G$ is gamma-distributed with density function
$$ p(g\mid x) = {1\over (x/m)^m\Gamma(m)}g^{m-1}e^{-gm/x}. $$Let $G=xV$. Then it follows that $V$ has the density
$$ p(v) = {m^m\over\Gamma(m)}v^{m-1}e^{-vm}. $$Therefore, in terms of the observed pixel intensities $g$ (realizations of $G$), we can write
$$ g = x v, $$where v is distributed as above, and has mean 1 and variance $1/m$.
Because of this special multiplicative noise nature of speckle, conventional smoothing filters are not particularly suitable as an aid to SAR image interpretation.
The gamma maximum a posteriori (gamma MAP) de-speckling filter may be derived from Bayes' Theorem. The a posteriori conditional probability for $x$, given intensity measurement $g$ is
$$ {\rm pr}(x\mid g) = { p(g\mid x){\rm pr}(x)\over p(g) }, $$where $p(g\mid x)$ is given above, ${\rm pr}(x)$ is the prior probability for $x$ and $p(g)$ is the total probability density for $g$. This formulation allows us to include prior knowledge of the signal statistics (or texture) if available. An empirical statistical model for $x$ is suggested by measurements of backscatter from ocean waves, namely
$$ {\rm pr}(x) \sim \left({\alpha\over\mu}\right)^\alpha {x^{\alpha-1}\over\Gamma(\alpha)}e^{-\alpha x/\mu}. $$This is just the gamma probability density with $\beta=\mu/\alpha$, and hence with mean $\alpha\beta= \mu$ and variance
$$ {\rm var}(x) = \alpha\beta^2 = \mu^2/\alpha. $$The parameters $\mu$ and $\alpha$ can be estimated as follows. By passing an $n\times n$ window over the image we can obtain $\bar g = \langle g\rangle$ and ${\rm var}(g)$. Then the estimates are
$$ \hat\mu = \bar g, $$and $$ \hat\alpha = {\hat\mu^2\over {\rm var}(x)} = {\bar g^2\over {\rm var}(x)} ={1 + 1/m \over {{\rm var}(g)/\bar g^2 - 1/ m}}. $$
Hence the posterior probability for $x$ given measurement $g$ is
$$ {\rm pr}(x\mid g) \sim {1\over (x/m)^m\Gamma(m)}g^{m-1}e^{-gm/x}\left({\alpha\over\mu}\right)^\alpha {x^{\alpha-1}\over\Gamma(\alpha)}e^{-\alpha x/\mu} =: L $$Taking the logarithm,
$$ \log L =\ m\log m -m\log x +(m-1)\log g - \log\Gamma(m)- mg/x +\alpha\log\alpha - \alpha\log\mu + (\alpha-1)\log x - \log\Gamma(\alpha) -\alpha x/\mu. $$We now get the maximum a posteriori (MAP) value for $x$ given the observed pixel intensity $g$ by maximizing $\log L$ with respect to $x$:
$$ {d\over dx}\log L = -m/x + mg/ x^2 + (\alpha-1)/x - \alpha/\mu = 0. $$This leads to a quadratic equation for the most probable signal intensity $x$,
$$ {\alpha\over\mu}x^2 + (m+1-\alpha)x - mg = 0, $$where the parameters $\mu$ and $\alpha$ are estimated locally. Note that in homogeneous regions where $m\approx \bar g^2/{\rm var}(g)$, $\hat\alpha\to\infty$. In that case $x\approx \hat\mu = \bar g$.
The gamma MAP filter is not appropriate to the complex off-diagonal matrix elements as their {\it a priori} statistics are not well understood. The Python script gamma_filter.py takes as input a polSAR image in covariance or coherency matrix form and filters the diagonal elements only:
run gamma_filter -h
If the flag -p is set, the filter will be run in parallel on the diagonal elements. This assumes that at least three IPython engines have been started, e.g., with the shell command
ipcluster start -n 3
run gamma_filter -d [400,300,400,400] \
/home/imagery/RS2_OK5491_PK71074_DK68879_FQ21_20100426_172459_HH_VV_HV_VH_SLC_MapReady/T3/polSAR.tif 12
run gamma_filter -p -d [400,300,400,400] \
/home/imagery/RS2_OK5491_PK71074_DK68879_FQ21_20100426_172459_HH_VV_HV_VH_SLC_MapReady/T3//polSAR.tif 12
We get a speedup of about a factor two. Here is the filtered image (left) compared to the original (right):
run /home/dispms -p [1,2,3] -P [1,6,9] \
-f /home/imagery/RS2_OK5491_PK71074_DK68879_FQ21_20100426_172459_HH_VV_HV_VH_SLC_MapReady/T3/polSAR_gamma.tif \
-F /home/imagery/RS2_OK5491_PK71074_DK68879_FQ21_20100426_172459_HH_VV_HV_VH_SLC_MapReady/T3/polSAR.tif -D [400,300,400,400]
